27 research outputs found
A refined Razumov-Stroganov conjecture II
We extend a previous conjecture [cond-mat/0407477] relating the
Perron-Frobenius eigenvector of the monodromy matrix of the O(1) loop model to
refined numbers of alternating sign matrices. By considering the O(1) loop
model on a semi-infinite cylinder with dislocations, we obtain the generating
function for alternating sign matrices with prescribed positions of 1's on
their top and bottom rows. This seems to indicate a deep correspondence between
observables in both models.Comment: 21 pages, 10 figures (3 in text), uses lanlmac, hyperbasics and epsf
macro
A refined Razumov-Stroganov conjecture
We extend the Razumov-Stroganov conjecture relating the groundstate of the
O(1) spin chain to alternating sign matrices, by relating the groundstate of
the monodromy matrix of the O(1) model to the so-called refined alternating
sign matrices, i.e. with prescribed configuration of their first row, as well
as to refined fully-packed loop configurations on a square grid, keeping track
both of the loop connectivity and of the configuration of their top row. We
also conjecture a direct relation between this groundstate and refined totally
symmetric self-complementary plane partitions, namely, in their formulation as
sets of non-intersecting lattice paths, with prescribed last steps of all
paths.Comment: 20 pages, 4 figures, uses epsf and harvmac macros a few typos
correcte
On FPL configurations with four sets of nested arches
The problem of counting the number of Fully Packed Loop (FPL) configurations
with four sets of a,b,c,d nested arches is addressed. It is shown that it may
be expressed as the problem of enumeration of tilings of a domain of the
triangular lattice with a conic singularity. After reexpression in terms of
non-intersecting lines, the Lindstr\"om-Gessel-Viennot theorem leads to a
formula as a sum of determinants. This is made quite explicit when
min(a,b,c,d)=1 or 2. We also find a compact determinant formula which generates
the numbers of configurations with b=d.Comment: 22 pages, TeX, 16 figures; a new formula for a generating function
adde
Finite-size left-passage probability in percolation
We obtain an exact finite-size expression for the probability that a
percolation hull will touch the boundary, on a strip of finite width. Our
calculation is based on the q-deformed Knizhnik--Zamolodchikov approach, and
the results are expressed in terms of symplectic characters. In the large size
limit, we recover the scaling behaviour predicted by Schramm's left-passage
formula. We also derive a general relation between the left-passage probability
in the Fortuin--Kasteleyn cluster model and the magnetisation profile in the
open XXZ chain with diagonal, complex boundary terms.Comment: 21 pages, 8 figure
Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon
The raise and peel model of a one-dimensional fluctuating interface (model A)
is extended by considering one source (model B) or two sources (model C) at the
boundaries. The Hamiltonians describing the three processes have, in the
thermodynamic limit, spectra given by conformal field theory. The probability
of the different configurations in the stationary states of the three models
are not only related but have interesting combinatorial properties. We show
that by extending Pascal's triangle (which gives solutions to linear relations
in terms of integer numbers), to an hexagon, one obtains integer solutions of
bilinear relations. These solutions give not only the weights of the various
configurations in the three models but also give an insight to the connections
between the probability distributions in the stationary states of the three
models. Interestingly enough, Pascal's hexagon also gives solutions to a
Hirota's difference equation.Comment: 33 pages, an abstract and an introduction are rewritten, few
references are adde
Inhomogeneous loop models with open boundaries
We consider the crossing and non-crossing O(1) dense loop models on a
semi-infinite strip, with inhomogeneities (spectral parameters) that preserve
the integrability. We compute the components of the ground state vector and
obtain a closed expression for their sum, in the form of Pfaffian and
determinantal formulas.Comment: 42 pages, 31 figures, minor corrections, references correcte
Exact expressions for correlations in the ground state of the dense O(1) loop model
Conjectures for analytical expressions for correlations in the dense O
loop model on semi infinite square lattices are given. We have obtained these
results for four types of boundary conditions. Periodic and reflecting boundary
conditions have been considered before. We give many new conjectures for these
two cases and review some of the existing results. We also consider boundaries
on which loops can end. We call such boundaries ''open''. We have obtained
expressions for correlations when both boundaries are open, and one is open and
the other one is reflecting. Also, we formulate a conjecture relating the
ground state of the model with open boundaries to Fully Packed Loop models on a
finite square grid. We also review earlier obtained results about this relation
for the three other types of boundary conditions. Finally, we construct a
mapping between the ground state of the dense O loop model and the XXZ
spin chain for the different types of boundary conditions.Comment: 25 pages, version accepted by JSTA
The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is
In 2010, Duminil-Copin and Smirnov proved a long-standing conjecture of
Nienhuis, made in 1982, that the growth constant of self-avoiding walks on the
hexagonal (a.k.a. honeycomb) lattice is A key identity
used in that proof was later generalised by Smirnov so as to apply to a general
O(n) loop model with (the case corresponding to SAWs).
We modify this model by restricting to a half-plane and introducing a surface
fugacity associated with boundary sites (also called surface sites), and
obtain a generalisation of Smirnov's identity. The critical value of the
surface fugacity was conjectured by Batchelor and Yung in 1995 to be This value plays a crucial role in our generalized
identity, just as the value of growth constant did in Smirnov's identity.
For the case , corresponding to \saws\ interacting with a surface, we
prove the conjectured value of the critical surface fugacity. A crucial part of
the proof involves demonstrating that the generating function of self-avoiding
bridges of height , taken at its critical point , tends to 0 as
increases, as predicted from SLE theory.Comment: Major revision, references updated, 25 pages, 13 figure
Temperley-Lieb Stochastic Processes
We discuss one-dimensional stochastic processes defined through the
Temperley-Lieb algebra related to the Q=1 Potts model. For various boundary
conditions, we formulate a conjecture relating the probability distribution
which describes the stationary state, to the enumeration of a symmetry class of
alternating sign matrices, objects that have received much attention in
combinatorics.Comment: 9 pages LaTeX, 11 Postscript figures, minor change